HIGH-PERFORMANCE SIGNAL ACQUISITION ALGORITHMS

FOR WIRELESS COMMUNICATIONS RECEIVERS

A Dissertation

by

KAI SHI

Submitted to the Office of Graduate Studies ofTexas A&M University

in partial fulfillment of the requirements for the degree of

DOCTOR OF PHILOSOPHY

August 2005

Major Subject: Electrical Engineering

HIGH-PERFORMANCE SIGNAL ACQUISITION ALGORITHMS

FOR WIRELESS COMMUNICATIONS RECEIVERS

A Dissertation

by

KAI SHI

Submitted to the Office of Graduate Studies ofTexas A&M University

in partial fulfillment of the requirements for the degree of

DOCTOR OF PHILOSOPHY

Approved by:

Chair of Committee, Erchin SerpedinCommittee Members, Scott L. Miller

Xi ZhangAndreas Klappenecker

Head of Department, Chanan Singh

August 2005

Major Subject: Electrical Engineering

iii

ABSTRACT

High-Performance Signal Acquisition Algorithms

for Wireless Communications Receivers. (August 2005)

Kai Shi, B.Sc., Nanjing University, Nanjing, China;

M.Eng., Southeast University, Nanjing, China

Chair of Advisory Committee: Dr. Erchin Serpedin

Due to the uncertainties introduced by the propagation channel, and RF and

mixed signal circuits imperfections, digital communication receivers require efficient

and robust signal acquisition algorithms for timing and carrier recovery, and interfer-

ence rejection.

The main theme of this work is the development of efficient and robust signal

synchronization and interference rejection schemes for narrowband, wideband and

ultra wideband communications systems. A series of novel signal acquisition schemes

together with their performance analysis and comparisons with existing state-of-the-

art results are introduced. The design effort is first focused on narrowband systems,

and then on wideband and ultra wideband systems.

For single carrier modulated narrowband systems, it is found that conventional

timing recovery schemes present low efficiency, e.g., certain feedback timing recov-

ery schemes exhibit the so-called hang-up phenomenon, while another class of blind

feedforward timing recovery schemes presents large self-noise. Based on a general re-

search framework, we propose new anti-hangup algorithms and prefiltering techniques

to speed up the feedback timing recovery and reduce the self-noise of feedforward tim-

ing estimators, respectively.

Orthogonal frequency division multiplexing (OFDM) technique is well suited for

wideband wireless systems. However, OFDM receivers require high performance car-

iv

rier and timing synchronization. A new coarse synchronization scheme is proposed for

efficient carrier frequency offset and timing acquisition. Also, a novel highly accurate

decision-directed algorithm is proposed to track and compensate the residual phase

and timing errors after the coarse synchronization step. Both theoretical analysis

and computer simulations indicate that the proposed algorithms greatly improve the

performance of OFDM receivers.

The results of an in-depth study show that a narrowband interference (NBI) could

cause serious performance loss in multiband OFDM based ultra-wideband (UWB) sys-

tems. A novel NBI mitigation scheme, based on a digital NBI detector and adaptive

analog notch filter bank, is proposed to reduce the effects of NBI in UWB systems.

Simulation results show that the proposed NBI mitigation scheme improves signifi-

cantly the performance of a standard UWB receiver (this improvement manifests as

a signal-to-noise ratio (SNR) gain of 9 dB).

v

To my parents and wife

vi

ACKNOWLEDGMENTS

I would like to thank my advisor, Professor Erchin Serpedin, for his constant

support, guidance and inspiration during my years at TAMU. This research work

would never have been accomplished without his encouragement and help.

Special thanks goes to Prof. Scott L. Miller for teaching me modulation theory

and serving as a member of my Ph.D. committee. I would also like to thank Prof.

Xi Zhang and Prof. Andreas Klappenecker for serving on my advisory committee. I

am also thankful to some professors in our department, Prof. Krishna R. Narayanan

for his lectures on coding theory, and Prof. Aydin Karsilayan for his inspiration and

guidance when I was working on the UWB project.

I also enjoyed working with many colleagues in the Wireless Communications

Laboratory (WCL), and in particular I would like to thank Dr. Yang Wang, Yik-

chung Wu, Dr. Zhongmin Liu, Dr. Shengjie Zhao, Dr. Jing Li, Dr. Guoshen Yue,

Nitin A. Nangare, Janath Peiris, Dr. Vivek Gulati, Hari Sankar, Yong Sun and Jun

Zheng. I owe also special thanks to some researchers from the Analog and Mixed

Signal Group at TAMU: Burak Kelleci, Timothy Fischer, and Haitao Tong.

I would like to thank Dr. Zoran Zvonar, Dr. Aiguo Yan, Deepak Mathew and

Lidwine Martinot for their encouragement and guidance when I worked as a system

engineering intern at Analog Devices. This successful experience will have a strong

impact on my future career.

Finally, I would like to express my gratitude to my parents and sister for their

encouragement and constant support. This dissertation is dedicated to my wife,

Likun Cao. Her love, understanding and encouragement represented the momentum

to complete this Ph.D. program.

vii

TABLE OF CONTENTS

CHAPTER Page

I INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . 1

A. Background . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1. Timing Recovery . . . . . . . . . . . . . . . . . . . . . 3

2. Carrier Recovery . . . . . . . . . . . . . . . . . . . . . 4

3. Interference Mitigation . . . . . . . . . . . . . . . . . 5

B. Organization of Dissertation . . . . . . . . . . . . . . . . . 5

II FAST FEEDBACK TIMING RECOVERY FOR NARROW-

BAND SYSTEMS . . . . . . . . . . . . . . . . . . . . . . . . . . 9

A. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 9

B. Signal Model and Feedback Timing Recovery . . . . . . . . 10

C. Hangup in Digital PLL . . . . . . . . . . . . . . . . . . . . 12

D. Anti-hangup Timing Recovery Scheme . . . . . . . . . . . 14

E. Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

III JITTER-FREE FEEDFORWARD TIMING RECOVERY FOR

NARROWBAND SYSTEMS . . . . . . . . . . . . . . . . . . . . 25

A. Jitter-Free Prefilter for Feedforward Timing Recovery

of Linear Modulations . . . . . . . . . . . . . . . . . . . . 25

1. Introduction . . . . . . . . . . . . . . . . . . . . . . . 25

2. Signal Model and Symbol Timing Estimators . . . . . 26

3. Nearly Jitter-Free Prefilter . . . . . . . . . . . . . . . 28

4. Summary . . . . . . . . . . . . . . . . . . . . . . . . . 34

B. Jitter-Free Prefilter for Feedforward Timing Recovery

of GMSK Modulations . . . . . . . . . . . . . . . . . . . . 34

1. Introduction . . . . . . . . . . . . . . . . . . . . . . . 34

2. Signal Model and Estimators . . . . . . . . . . . . . . 35

3. Self-Noise Compensating Prefilter . . . . . . . . . . . 39

4. Summary . . . . . . . . . . . . . . . . . . . . . . . . . 42

IV EFFICIENT COARSE FRAME AND CARRIER SYNCHRO-

NIZATION FOR OFDM SYSTEMS . . . . . . . . . . . . . . . . 43

A. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 43

viii

CHAPTER Page

B. Signal Model . . . . . . . . . . . . . . . . . . . . . . . . . 45

C. Maximum Likelihood Estimator . . . . . . . . . . . . . . . 47

D. Theoretical Analysis of Estimators . . . . . . . . . . . . . 56

1. Coarse Packet Detection . . . . . . . . . . . . . . . . . 56

2. Fine Packet Detection . . . . . . . . . . . . . . . . . . 58

3. Carrier Frequency Offset Estimation . . . . . . . . . . 60

E. Numerical Analysis of Estimators . . . . . . . . . . . . . . 63

1. Performance of Algorithms . . . . . . . . . . . . . . . 65

2. System Performance . . . . . . . . . . . . . . . . . . . 69

F. Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

V DECISION-DIRECTED FINE SYNCHRONIZATION FOR

OFDM SYSTEMS . . . . . . . . . . . . . . . . . . . . . . . . . . 71

A. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 71

B. Signal Models . . . . . . . . . . . . . . . . . . . . . . . . . 71

C. Data-aided Estimator . . . . . . . . . . . . . . . . . . . . . 73

D. Proposed Synchronization Scheme . . . . . . . . . . . . . . 74

1. Decision-Directed Estimator . . . . . . . . . . . . . . 74

2. Closed-Loop Scheme . . . . . . . . . . . . . . . . . . . 77

E. Computer Simulations . . . . . . . . . . . . . . . . . . . . 78

F. Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

VI ADAPTIVE NARROWBAND INTERFERENCE MITIGA-

TION FOR MB-OFDM UWB SYSTEMS . . . . . . . . . . . . . 80

A. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 80

B. Signal Model . . . . . . . . . . . . . . . . . . . . . . . . . 83

1. Single-Band OFDM Systems . . . . . . . . . . . . . . 83

2. Multi-Band OFDM Systems . . . . . . . . . . . . . . 85

C. Impact of Narrowband Interference . . . . . . . . . . . . . 86

1. Optimization of Data Converters . . . . . . . . . . . . 86

2. Carrier Synchronization . . . . . . . . . . . . . . . . . 92

3. System Performance of a MB-OFDM Receiver . . . . 96

D. Narrowband Interference Suppression . . . . . . . . . . . . 98

1. Digital NBI Detection and Mitigation . . . . . . . . . 99

2. Analog NBI Cancellation . . . . . . . . . . . . . . . . 102

3. Performance of the Mixed NBI Suppression Scheme . 111

E. Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . 113

VII CONCLUSIONS . . . . . . . . . . . . . . . . . . . . . . . . . . . 115

ix

CHAPTER Page

REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116

APPENDIX A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126

APPENDIX B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130

APPENDIX C . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132

APPENDIX D . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133

APPENDIX E . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135

APPENDIX F . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137

VITA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139

x

LIST OF TABLES

TABLE Page

I Optimum Ω versus ADC bit number for OFDM receiver in AWGN

channels. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

II The simulation configuration of MB-OFDM systems. . . . . . . . . . 98

xi

LIST OF FIGURES

FIGURE Page

1 Signal acquisition at digital receiver. . . . . . . . . . . . . . . . . . . 2

2 Feedback timing recovery scheme. . . . . . . . . . . . . . . . . . . . . 3

3 Feedforward timing recovery scheme. . . . . . . . . . . . . . . . . . . 4

4 Coarse and fine synchronization of OFDM receiver. . . . . . . . . . . 5

5 Content and organization of dissertation. . . . . . . . . . . . . . . . . 6

6 Conventional timing recovery scheme. . . . . . . . . . . . . . . . . . 12

7 The S-curves of timing error detectors. . . . . . . . . . . . . . . . . . 14

8 Timing errors versus the number of symbols for Gardner’s TED. . . . 15

9 The shifted version of S-curves. . . . . . . . . . . . . . . . . . . . . . 16

10 The new S-curves. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

11 Fast timing recovery scheme. . . . . . . . . . . . . . . . . . . . . . . 17

12 Ph versus different L. . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

13 Performance comparison in AWGN channels, Es/No = 15 dB, 100 runs. 21

14 Performance comparison in flat fading channels, BLT = 0.01,

Es/No = 20 dB, fdT = 0.0002. . . . . . . . . . . . . . . . . . . . . . . 23

15 Plots of Gm(1, 0)/Gm(0, 0) for m = 1, 2 versus various roll-off

factors ρ. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

16 Comparison of symbol timing estimators for P = 2. . . . . . . . . . . 32

17 Comparison of the performance of estimator for different N . . . . . . 33

18 Feedforward clock recovery scheme. . . . . . . . . . . . . . . . . . . . 37

xii

FIGURE Page

19 Performance comparisons of various estimators for GMSK modulations. 38

20 Amplitude response of filters. . . . . . . . . . . . . . . . . . . . . . . 40

21 The performance of estimators with and without prefilter. . . . . . . 41

22 Mean values of timing metrics, SNR=20dB. . . . . . . . . . . . . . . 52

23 Coarse packet detection and fine packet detection(Ω = L). . . . . . . 55

24 Probability of Z(l) > 0 for (a) Es/N0 = 10dB, (b) Es/N0 = 40dB.

The solid and the dash lines are associated with the metrics of [+B

+B -B +B] and [+A +A], respectively. . . . . . . . . . . . . . . . . . 57

25 The probability of false alarm versus N and Tc. . . . . . . . . . . . . 59

26 MSE of carrier frequency offset estimator in AWGN channels. . . . . 61

27 Comparison of MSE and complexity of carrier frequency offset

estimators in AWGN channels, SNR=15dB. . . . . . . . . . . . . . . 64

28 Time offset estimation l̂ in frequency-selective channels. . . . . . . . 66

29 The probabilities of timing errors that will cause ISI in frequency-

selective channels. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

30 MSE of the carrier frequency offset estimators in frequency-selective

channels. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

31 BER performance versus SNR in frequency-selective channels. . . . . 69

32 The receiver structure. . . . . . . . . . . . . . . . . . . . . . . . . . . 75

33 Normalized MSE (normalized by 10−12) of open-loop DD SFO

estimators. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

34 Normalized MSE (normalized by 10−12) of closed-loop SFO estimators. 78

35 The signal model of single-band OFDM systems. . . . . . . . . . . . 83

36 γq versus Ω and number of bits of ADC. . . . . . . . . . . . . . . . . 88

xiii

FIGURE Page

37 SINR curves of ADC, Pc/Pn = 10dB. . . . . . . . . . . . . . . . . . . 91

38 Theoretical (dashed line) and simulated (solid line) optimum γqversus SIRs. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

39 SINR with non-optimal γq (solid line) and with optimal γq (dashed

line), Pc/Pn = 10dB. . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

40 The performance of CFO estimators versus SIR, v = 0.08, L =

128, and Pc/Pn = 5dB. . . . . . . . . . . . . . . . . . . . . . . . . . . 96

41 The block error rate versus Pc/Pn and SIR. . . . . . . . . . . . . . . 97

42 The magnitude-squared of FFT bins: |X[k]|2, SIR = 8 dB, Pc/Pn =5dB. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100

43 Performance comparison of MB-OFDM receiver assuming ideal

NBI detection. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102

44 The structure of proposed NBI suppression scheme. . . . . . . . . . . 103

45 The amplitude and step response of adaptive analog notch filter,

Anotch = −20dB, fBW = 24MHz. . . . . . . . . . . . . . . . . . . . . 104

46 Simulated BLER versus fBW of AANF. . . . . . . . . . . . . . . . . 105

47 Notch filter based on band-pass feed-forward cancelation. . . . . . . . 106

48 Circuit structure of notch filter with a single notching frequency,f0controlled by the digital word, W . . . . . . . . . . . . . . . . . . . . 106

49 AANF frequency response versus W , Anotch = −40dB. . . . . . . . . 108

50 AANF filter characteristics versus W , Anotch = −40dB. . . . . . . . . 109

51 The working flow of proposed NBI suppression scheme. . . . . . . . . 110

52 Time and frequency domain signals after AGC: without or with

adaptive analog notch filter, SIR=−10dB, fint = 20, Anotch =−20dB, fBW = 24MHz. . . . . . . . . . . . . . . . . . . . . . . . . . 112

xiv

FIGURE Page

53 Performance comparison of MB-OFDM receiver, ideal NBI detec-

tion, 200Mbps date rate, UWB channel model 1. . . . . . . . . . . . 114

1

CHAPTER I

INTRODUCTION

A. Background

Thanks to its convenience, wireless communication is a promising technology and is

becoming dominant among all last-one-mile technologies. To transmit data symbols

over the air efficiently, wireless transceivers often involve carrier modulation. In the

transmitter, after digital-to-analog (D/A) conversion, the complex baseband signals

are modulated to high frequency signals by I (in phase) and Q (quadrature) mod-

ulator. At the receiver, the received signals are converted into the baseband by IQ

demodulator. Then some optimal or sub-optimal detectors can be used for data re-

covery. Due to the RF imperfections and the uncertainty of propagation channel, the

received signals could be seriously distorted and interfered by other wireless systems.

As shown in Fig. 1, to compensate various distortions and interferences, the

digital receiver requires efficient carrier and timing recovery, channel estimation and

interference rejection schemes. This family of algorithms is generally referred to as

signal acquisition. Efficient and robust signal acquisition algorithms are very critical

for the success of data recovery. Note that, for some wireless systems, only some

blocks in Fig. 1 are required and their processing flow may be different from the order

shown in Fig. 1.

References [1] - [4] are excellent books on carrier and timing recovery that have

been reported in the literature. In addition, most of the existing results on interference

rejection were well summarized in [5]- [6] and their references. The goal of this

dissertation is to build a general framework for signal acquisition and explore high-

The journal model is IEEE Transactions on Automatic Control.

2

Digital Receiver

ModulatorIQ

D/A

ChannelPropagation

TransmitterBaseband

IQ DemodulatorA/D

InterferenceRejection

TimingRecovery

CarrierRecovery Detector

Data

ChannelEstimation

Fig. 1. Signal acquisition at digital receiver.

performance algorithms for various wireless systems. Note that channel estimation is

not the focus of this dissertation and some existing work on channel estimation can

be found in [1]- [2].

Depending on whether the receiver knows the transmitted data or not, the signal

acquisition schemes can be classified into three broad categories: data-aided (DA),

decision-directed (DD), and non-data aided (NDA) or blind. All these schemes will be

addressed in this dissertation, e.g., the DA scheme is used for coarse synchronization

of OFDM receivers (Chapter IV), the DD scheme is used for fine synchronization of

OFDM receivers (Chapter V), and NDA schemes are discussed in Chapters II and

III.

3

1. Timing Recovery

In digital communication receivers, the received signal is sampled by means of an

analog to digital (A/D) converter. However, there is an arbitrary symbol timing delay

between the receiver and transmitter, which is not known by the A/D converter. To

get correct data strobe for detection, the digital receiver requires timing recovery. At

first, the timing offset is estimated using the received data and then a timing error

controller is used to compensate the estimated timing offset. Depending on the type

of compensation method used, timing recovery schemes can be classified into two

classes: feedback and feedforward. In the feedback timing recovery scheme shown

Data Detector

Free RunningClock

A/D InterpolatorDecimator

Timing OffsetEstimator

Loop filter

To

Fig. 2. Feedback timing recovery scheme.

in Fig. 2, the estimated timing error is fed back to adjust the time offset of digital

interpolator (or A/D converter) and the resulting new samples are used to estimate

the residual timing error. Based on this iterative operation, the system timing error

converges to the stable operation point. Thanks to its simplicity, the feedback scheme

is often used in practice. However, for certain initial values of the timing error, the

acquisition time of blind feedback timing recovery schemes could be extremely large.

This problem is called hang-up [7]- [9] and will be addressed in Chapter II. As shown in

Fig. 3, a typical feedforward timing recovery scheme uses the estimated timing offset

4

To

Free RunningClock

A/D

Timing OffsetEstimator

InterpolatorDecimator Data Detector

Fig. 3. Feedforward timing recovery scheme.

to control the digital interpolator. Since there is no feedback between the timing

offset estimator and timing error controller, the hang-up phenomenon is avoided in

feedforward systems [10]- [15]. However, the feedforward timing estimator presents

the self-noise which is embedded in most of blind estimators [16].

2. Carrier Recovery

Due to the imperfections in oscillators and Doppler shifts introduced by the propaga-

tion channel, carrier frequency and phase offsets are present in the received signal. The

carrier synchronization task resumes to estimating and compensating these offsets at

the receiver. Compared to single carrier systems, orthogonal frequency division mul-

tiplexing (OFDM) systems [17] are very sensitive to carrier frequency offsets. Several

schemes have been proposed for coarse estimation of the carrier frequency offset [18]-

[26], which are not efficient in terms of performance and complexity. Therefore, an

important effort in this dissertation is allocated to the problem of carrier synchroniza-

tion for OFDM systems. For OFDM receivers, the performance of carrier frequency

offset estimator depends on the initial timing synchronization. Therefore, the timing

synchronization of OFDM receivers will be also explored in this work.

As shown in Fig. 4, the synchronization of OFDM systems consists of two steps:

coarse and fine synchronization. The coarse synchronization step assumes the task of

5

removing the large timing and carrier frequency offsets before the fast Fourier trans-

form (FFT) block. After coarse synchronization, the received signal is converted into

the frequency domain and might still present residual phase and timing errors. With-

out compensation, these residual errors could introduce significant performance loss.

Since these residual errors are small in magnitude, fine synchronization is normally

performed in the frequency domain using tracking.

Coarse Synchronization FFT Synchronization

Fine

Fig. 4. Coarse and fine synchronization of OFDM receiver.

3. Interference Mitigation

Ultra-wideband (UWB) [27] systems provide high data rate, low-power, huge spatial

capacity and high precision ranging for the quickly growing home networking market.

Since UWB systems use very low transmission power and operate at license-free

bands [28], a narrowband interference (NBI) might represent a very critical obstruc-

tion for the successful operation of UWB systems. It is found that the NBI could cause

serious performance loss for these systems. Conventional NBI mitigation schemes [5]-

[6] requires high precision ADC and are not feasible for UWB receivers. Designing

low complexity and robust NBI mitigation schemes for UWB receivers will represent

an additional major research thrust of this work.

B. Organization of Dissertation

The unifying feature of this work is the development of robust and efficient signal

acquisition and interference cancellation schemes for general narrowband, wideband

6

and ultra wideband communication systems. Novel and high performance signal

synchronization schemes are first proposed for narrowband communication systems

in Chapters II and III of this work. In Chapters IV and V, the analysis is focused on

the important problem of designing robust and efficient signal acquisition algorithms

for the class of wideband systems belonging to the OFDM family. Finally, Chapter VI

explores the applicability and robustness of the wideband signal interception schemes

to ultra wideband systems. Also, Chapter VI proposes a novel, robust and high

performance narrowband interference mitigation scheme for the family of multiband

OFDM-based UWB transceivers.

All the proposed signal acquisition and interference mitigation schemes are in-

troduced gradually starting with narrowband systems, then extending the analysis

to wideband and ultra wideband systems. Fig. 5 illustrates how this dissertation is

structured.

Narrowband

carrier

modulated

systems

Single

modulated

systems

carrier

frequencyHighly

selective

selectiveFrequency

Flat−fadingAWGN or

Coarse synchronization (time domain)

Fine synchronization(frequency domain)

Narrowbandinterference mitigation

Chapter III(feedforward)

Timing Recovery

(feedback)Timing Recovery

Chapter II

Chapter IV

Chapter VI

Chapter V

Ultra−wideband

System bandwidth Propagation channel

WidebandMultiple

Fig. 5. Content and organization of dissertation.

In Chapters II and III, we will focus on exploring the high-performance timing

7

recovery schemes for narrowband systems, where the wireless propagation channel

can be assumed to be AWGN channel or flat fading channel.

Chapter II deals with the well known hang-up problem for blind feedback timing

recovery schemes. When the initial timing offset of blind feedback timing recovery

scheme is close to half of symbol period, the acquisition time becomes extremely long.

A novel low complexity anti-hangup algorithm will be proposed to resolve the hang-

up problem. Simulation results show that the proposed algorithm greatly reduces the

probability of hang-up and speeds up the feedback timing recovery.

The performance of blind feedforward timing recovery scheme will be analyzed in

Chapter III. It is found that the feedforward timing recovery schemes exhibit a large

self-noise especially for systems with small roll-off factors. The statistical analysis

of self-noise leads to a novel prefiltering scheme, which can be used to reduce the

self-noise of blind feedforward timing estimators.

Chapter IV, V and VI deal with high performance synchronization and inter-

ference mitigation algorithms for wideband and ultra-wideband systems, where the

propagation channel could be highly frequency-selective.

Chapter IV deals with coarse carrier and timing synchronization for OFDM re-

ceivers. A novel and robust synchronization scheme is proposed for efficient carrier

frequency offset and timing acquisition.

Fine synchronization (tracking) in frequency domain is discussed in Chapter V.

A new decision directed algorithm is proposed to track the residual phase and timing

error. Both theoretical analysis and computer simulations indicate that the pro-

posed algorithms greatly improve the efficiency and performance of standard OFDM

receivers.

Chapter VI is dedicated to the problem of narrowband interference mitigation

in multiband OFDM (MB-OFDM) UWB systems. A new NBI cancellation scheme,

8

based on a digital NBI detector and adaptive analog notch filter bank, is proposed to

reduce the effects of NBI on UWB systems. Simulation results show that the proposed

scheme improves significantly the performance of UWB receivers, an improvement

which translates into an SNR gain of 9 dB. It should be noted that some research

results on the adaptive analog notch filtering were contributed in collaboration with

other researchers: Burak Kelleci, Timothy W. Fischer and Dr. Aydın İlker Karsilayan.

In fact, their contributions proved to be of fundamental importance for this study,

and of very substantial value.

Some concluding remarks are provided in Chapter VII. Appendices A-F present

some mathematical results that are used in the previous chapters.

9

CHAPTER II

FAST FEEDBACK TIMING RECOVERY FOR NARROWBAND SYSTEMS

A. Introduction

Thanks to its simplicity and robust tracking capability, the digital phase-lock loop

(PLL) is often used for timing recovery [4]. Normally, the digital PLL for timing

recovery consists of three elements, which include a timing error detector to estimate

the current timing error, a lowpass loop filter to decrease the estimation error and a

timing corrector to control the timing error.

Fast timing acquisition and good tracking performance are desirable but are

difficult to achieve simultaneously because the design of PLL is a trade-off between

acquisition time and tracking error. In noise-free channels, as long as the loop filter

is determined, the average acquisition time should be a constant. However, for some

timing error detectors [29]- [31], the PLL occasionally presents a very long acquisition

time, a phenomenon which is referred to as the hangup problem [7]- [9]. The slow

timing recovery is undesirable especially for burst transmission systems.

By introducing a hysteresis effect using a special preamble sequence, the au-

thors of [32] proposed a fast timing recovery scheme to avoid the hangup in the

decision-directed timing synchronizer [29]. However, this method works only for

partial-response signals. The hangup problem also exists in blind timing synchro-

nizers [30]- [31] for linear and non-linear modulations and it requires a more general

approach.

To resolve this problem, in this chapter we propose a novel two-step hangup

robust timing recovery scheme. Based on an initial estimate obtained in the first

step, we try to reduce the timing uncertainty from [−T/2, T/2] to [−T/4, T/4] (T is

10

the symbol period), and thus to avoid the hangup in the second step. The increased

complexity is very low since the proposed scheme can use the same timing error

detector as that used in the conventional scheme. We provide simulation results for

two well-known timing error detectors [30]- [31] and show that the proposed scheme

greatly speeds up the timing recovery for both linearly and nonlinearly modulated

transmissions [1].

B. Signal Model and Feedback Timing Recovery

In AWGN channels, the received signal can be expressed by

r(t) = x(t) + n(t) , (2.1)

where x(t) represents the transmitted linearly or non-linearly modulated signal and

n(t) denotes the band-limited additive noise. In (2.1), we omit carrier frequency offset

since timing synchronizer to be discussed is robust to such small frequency offset.

After passing through matched filter, the received signal becomes y(t). At times

lT − ǫT , the sampled signal can be expressed as

yl(ǫ) = y(lT − ǫT ) , (2.2)

where T , l and ǫT represent the data symbol period, the data symbol index and the

fractional timing offset in the analog to digital converter (ADC), respectively. For

simplicity, we omit the carrier frequency offset and phase offset in (2.2) since the

timing estimators we will use later are robust to these offsets.

The standard digital phase-lock loop (PLL) is often used for timing recovery.

The digital PLL for timing recovery is usually made up of three elements [1]- [2]: a

timing error detector (TED), a loop filter and a timing corrector. First, the TED

11

outputs a symbol-rate error signal e(l), which depends on the time estimates {ǫ̂l}.

Reference [30] provides the derivation of a lot of timing error detectors which can

be blind, decision directed or data-aided. In this chapter, we focus on blind timing

error detectors. Note that the PN sequence synchronization for the spread spectrum

receiver is out of the interests of this chapter. For linearly modulated signals, the

Gardner’s TED [30] can be used

e(l) = Re{

y∗l−1/2(ǫ̂l−1) [yl−1(ǫ̂l−1) − yl(ǫ̂l)]}

. (2.3)

For minimum shift keying (MSK) type signals, the Mengali’s TED [31] and [1] can

be used

e(l) := e(l, D) = (−1)D+1Re{

y2l−1/4(ǫ̂l−1)y∗2l−D−1/4(ǫ̂l−D−1)

−y2l+1/4(ǫ̂l)y∗2l−D+1/4(ǫ̂l−D)}

(2.4)

In (2.4), D is a design parameter and should be a positive integer. It is pointed out

in [1] that D = 1 is preferable for MSK and D = 2 leads to the best performance for

Gaussian MSK (GMSK). From the above equation, the sampling clock for Gardner’s

TED and Mengali’ TED should be at least Ts = T/2 and Ts = T/4, respectively. The

error signal is used to recursively update the timing estimates in the loop filter. For

the first-order loop filter, the updating equation is given by:

ǫ̂l = ǫ̂l−1 − γe(l) , (2.5)

where γ denotes the step size.

Finally, the timing estimates ǫ̂l are exploited to control the timing corrector. De-

pending on the sampling scheme, the timing correction can be realized with different

methods. If synchronous sampling is used, the timing estimates are fed into a number

12

controlled oscillator (NCO) to adjust the sampling clock in the ADC. If asynchronous

sampling is used, as shown in Fig. 6, the sampling clock in ADC is free running and

the timing errors are corrected by a digital timing interpolator [33]- [34]. Since the

asynchronous sampling easily leads to high performance all-digital receivers, we will

focus on asynchronous sampling hereafter. It should be also mentioned that the pro-

posed scheme works for synchronous sampling. In Fig. 6, after timing acquisition is

obtained, the decimator outputs symbol-rate samples to data detector. For linearly

modulated systems, the lowpass filter can be a matched filter while for nonlinearly

modulated systems, an anti-alias lowpass filter should be used.

ADC Interpolator

Loop Filter

Decimator

FilterMatched Timing Error

Detector

To

Detector

ClockFree−running

r(t) y(t) yl(ǫl)

ǫ̂l

yl(ǫ̂)

e(l)

Fig. 6. Conventional timing recovery scheme.

C. Hangup in Digital PLL

The design of a loop filter is in general a trade-off between acquisition timing and

tracking performance. In particular, for the first-order loop filter, if the step-size γ is

reduced, the PLL needs a longer acquisition time, while the tracking error becomes

smaller.

The S-curve of the timing error detector is often used to investigate the per-

formance of PLL and is defined by the conditional expectation of the timing error

13

signal:

S(δ) = E{e(l)|δ} , (2.6)

where δ is the difference between the true time offset and the current timing estimate

δ = (ǫ̂l−ǫl)T . It is the S-curve that determinates the strength of pulling the PLL from

the initial timing offset to the stable point δ = 0. Also, the acquisition and tracking

performance depends on the loop filter. From [1], the loop bandwidth during tracking

is given by

BLT =γA

4 − 2γA , (2.7)

where A is the slope of S-curve at δ = 0 and γA < 2 is assumed. In noise-free channels,

the acquisition time of the digital PLL can be approximated by (pp. 403-404, [2])

Tacq ≈2|ǫ0|BL

, (2.8)

which depends on the loop bandwidth and initial timing offset ǫ0. Assuming ǫ′0s are

uniformly distributed over [−1/2, 1/2], we obtain the average acquisition time

T̄acq =1

2BL=

(2 − γA)TA

. (2.9)

Therefore, in noise-free channels, the acquisition time of the digital PLL is often

assumed to be inversely proportional to the loop bandwidth BL. However, for some

timing recovery schemes, the digital PLL occasionally requires a very long acquisition

time. This phenomenon is called hangup and often happens in PLLs [7]- [9].

In Fig. 7, we plot the S-curves of different timing error detectors in noise-free

channels. For Gardner’s TED, we assume that the system uses a QPSK modulation

and a raised cosine (RC) filter with roll-off factor ρ = 0.5. For Mengali’s TED, a

GMSK modulated system with pre-modulation bandwidth BT = 0.3 is assumed.

These simulation parameters will be used throughout this chapter.

14

−0.5 0 0.5−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

S−

curv

es

Gardner’s TED, ρ=0.5Mengali’s TED, BT=0.3

δ/T

Fig. 7. The S-curves of timing error detectors.

The above S-curves have nulls at δ = 0, which corresponds to the desirable stable

point. However, the S-curves have additional zeros at δ = ±T/2. Therefore, if the

initial timing offset is close to ±T/2, the average output of TED will be very small.

Due to the small pull-in strength at δ = ±T/2, it often takes the PLL a very long

time to converge to the stable point.

In Fig. 8, the Gardner’ TED is used and we plot the time errors versus the number

of symbols for different initial timing offsets ǫ0. In the case of ǫ0T = 0.45T , the PLL

presents monotonic convergence. However, when ǫ0T = 0.5T , the PLL fluctuates

around 0.5T for a long time before it begins to converge toward the stable point.

D. Anti-hangup Timing Recovery Scheme

To remedy the hangup problem, one may use different loop filters during acquisition

and tracking (pp. 405-406, [2]), e.g., to obtain fast acquisition, one may use a larger

step size, which is replaced by a smaller step size to reduce the mean-square error

during tracking. It is found that the probability of hangup is greatly reduced (but

15

0 100 200 300 400 500 600 700 800 900−0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

The number of symbols

δ/T

BLT=0.005

ε0=0.5T

ε0=0.45T

Fig. 8. Timing errors versus the number of symbols for Gardner’s TED.

not avoided) by this method.

From the discussion in Section III, we know that the hangup will happen only

if the initial timing offset is close to ±T/2. For discussion purpose, we divide the

whole range of timing uncertainty into two subsets: t1 = [−T/4, T/4] and t2 =

(T/4, T/2] ∪ [−T/2,−T/4). If the subset of initial timing error is known in the

receiver, we may be able to avoid hangup by proper controlling. In the timing error

detector, shifting the samples by T/4 in (2.6), we obtain the shifted S-curves:

Ss(δ) = E{e(l − 1/4)|δ} . (2.10)

As shown in Fig. 9, the shifted S-curves are even functions and cross the zero axis at

δ = ±T/4. Also, Ss(ǫ0) can be estimated as follows

S1 =1

L

L−1∑

l=0

e(l − 1/4) , (2.11)

where we assume that the timing corrector and loop filter are not working during

16

−0.5 0 0.5−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

Ss(

δ)

Gardner’s TEDMengali’s TED

δ/T

Fig. 9. The shifted version of S-curves.

t < LT and ǫ̂l = 0 for l < L. Thus, the subset of initial timing offset may be easily

detected as follows

ǫ0T ∈ t1 if S1 ≥ 0 or ǫ0T ∈ t2 if S1 < 0 . (2.12)

If we detect ǫ0T ∈ t1, e(l) should be used to update the timing estimates in the

loop filter t > LT and the decimator outputs yl(ǫ̂l) for data detection. However, if

ǫ0T ∈ t2 is detected, to avoid hangup, we should use e(l − 1/2) to control the loop

filter. Assuming the coarse detections are always correct, we obtain the new S-curves

in Fig. 10. Besides the stable point at δ = 0, there are two additional stable points

at δ = ±T/2. If ǫ0T ∈ t1, the PLL will converge to δ = 0. However, if ǫ0T ∈ t2, it

will converge to δ = ±T/2. Thus, to obtain correct data detection, yl±1/2(ǫ̂l) should

be used in the decimator for ǫ0 ∈ t2.

Based on above results, in Fig. 11, we propose a new fast timing recovery scheme

which assumes a two-step operation:

17

−0.5 0 0.5−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

δ/T

New

S−

curv

es

Gardner’s TEDMengali’s TED

Fig. 10. The new S-curves.

ADC Interpolator

Decimator

FilterMatched Anti−hangup

Controllor

Loop Filter

Timing ErrorDetector

To

Detector

ClockFree−running

r(t) y(t) yl(ǫl)

ǫ̂l

e(l)

e(l − 1/4)

l ≤ L

l > L

Fig. 11. Fast timing recovery scheme.

18

1. Feedforward estimation during t ≤ LT according to (2.11). The output of tim-

ing error detector is switched to the accumulator in the anti-hangup controller,

then the feedfoward estimation is used to obtain the coarse information (t1 or

t2) of ǫ0T using (2.12).

2. Feedback timing recovery during t > LT . Depending on the coarse timing

information obtained in the first step, the TED selects a proper set of samples

to estimate timing errors and update the loop filter for tracking.

The performance of the new scheme depends on the design parameter L. If L

is too small, the estimate S1 will be unreliable and will result in incorrect timing

detection, which may further lead to hangup. Assuming hangup will happen only if

δ0 = ±T/2, the conditional probability

Ph := Prob {[S1 > 0|ǫ0T = ±T/2] ∪ [S1 < 0|ǫ0T = 0]}

should be kept very low to avoid hangup. On the other hand, if in the first step we

choose L too large, the overall acquisition time may be increased compared to the

conventional scheme. Therefore, small L should be chosen as long as Ph is kept low.

Although the variance of e(l) in Gardner’s TED can be calculated as [35], it is found

that e(l) can not be approximated to be Gaussian distributed for nonzero δ. Thus, it

is very difficult to obtain the theoretical value of Ph. In Fig. 12, we plot the simulation

results of Ph versus different L′s for both Gardner’s TED and Mengali’s TED. For

Gardner’s TED with ρ = 0.5, L = 20 may be enough to keep Ph low. However, due to

its higher order (4th order) nonlinearity (2.4), Mengali’s TED does not perform well

at low SNR and requires a longer L. As pointed out in [36], averaging e(l, D) over

different values of D can improve the performance of feedforward estimation. Thus,

19

0 2 4 6 8 1010

−4

10−3

10−2

10−1

100

Es/N

o (dB)

Ph

L=20L=40

ρ=0.5, QPSK

(a) Gardner’s TED

4 6 8 10 12 14 16 1810

−4

10−3

10−2

10−1

100

Es/N

o (dB)

Ph

L=20, D=2L=20, CombiningL=40, D=2

GMSK, BT=0.3

(b) Mengali’s TED

Fig. 12. Ph versus different L.

20

in the first step operation, the following estimator can be used to reduce Ph:

S2 =1

3L

3∑

D=1

L−1∑

l=0

e(l − 1/4, D) . (2.13)

As shown in Fig. 12 (b), for the same L, the simulated Ph with the above combining

estimator is much lower than the Ph obtained with estimator (2.4).

To verify the advantage of the proposed timing recovery scheme, we run simula-

tions to compare the performance of the different timing recovery schemes. In simula-

tions, we assume a system with QPSK modulation and a RC filter with ρ = 0.5. For

conventional and proposed schemes, the step size of loop filter is fixed to γ = 0.013,

which leads to a loop bandwidth during tracking: BLT = 0.005. Also, L = 20 symbols

are used during the first step of the proposed scheme.

For the adaptive loop filter scheme, to speed up the acquisition, a larger step

size γ = 0.026 is used during initial acquisition. According to (pp. 405-406, [2]),

for Gardner’s TED, e(l − 1/4) can be used to detect if PLL is in lock. From [30],

denoting H(f) as the Fourier transform of RC filter, we can express the expectation

of e(l − 1/4) by

E[e(l − 1/4)] = cos(2πδ) · 2T

∫ 1/2T

−1/2TH(f − 1

2T)H(f +

1

2T) cos(πfT )df , (2.14)

which is maximum when the PLL is in lock. A low-pass filter is used to average

e(l − 1/4)

Ld(l) = γfLd(l − 1) + (1 − γf )e(l − 1/4) , (2.15)

where γf denotes the forgetting factor and our simulations assume γf = 0.95. Ld(l)

is then compared with an appropriate threshold Thr for lock detection. As long as

the PLL is in lock, the step size of loop filter will be changed to γ = 0.013 to reduce

the tracking error. A threshold of Thr = 0.11, which is a half of (2.14) when δ = 0

21

and ρ = 0.5, is used in simulations.

0 100 200 300 400 500 600 700 800 900

−0.5

−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

0.5

The number of symbols

δ/T

(a) Conventional timing recovery scheme

γ=0.013

0 100 200 300 400 500 600 700 800 900

−0.5

−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

0.5

The number of symbols

δ/T

(b) New timing recovery scheme

γ=0.013

Step 1

Step 2

Fig. 13. Performance comparison in AWGN channels, Es/No = 15 dB, 100 runs.

Fig. 13 shows 100 runs in AWGN channels, using different initial timing offsets

and noise sequences. Figs. 13 (a), (b) are the simulation results of the conventional

scheme, the proposed scheme, and adaptive loop filter scheme, respectively. Since all

schemes use the same loop bandwidth during tracking, they present similar tracking

22

error. However, their acquisition performance is very different. For the conventional

scheme, although all runs finally converge to δ = 0, sometimes the acquisition is very

slow. In 5 out of 100 runs, the convergence to δ = 0 is not achieved even after 300

symbols. On the other hand, the proposed scheme exhibits fast time recovery. For

all experiments using the proposed scheme, the convergence is obtained in less than

150 symbols.

Interesting enough, the timing acquisition of the proposed scheme is even faster

than that of conventional scheme when ǫ0T 6= ±T/2. This can be explained by the

smaller timing uncertainty obtained in the feedfoward estimation of proposed scheme.

In noise-free channels, assuming ǫ0 are uniformly distributed over [−T/4, T/4] after

feedforward estimation, we obtain the average acquisition time of proposed scheme

T̄acq =1

4BL+ LT , (2.16)

which is only a half of (2.9) if LT is omitted.

It is noticed that the proposed scheme presents three stable points (0,±T/2),

which verifies our previous discussion. However, correct data strobes can be obtained

by properly controlling the decimator in Fig. 11.

The performance of conventional and proposed timing recovery schemes is also

evaluated in flat fading channels. Due to the channel fading, the received signal

exhibits a large dynamic range, which may lead to a variable loop bandwidth. To

keep the loop bandwidth constant, before the signal enters the timing loop, a AGC is

used to normalize the received signal by using the root mean-square of the received

signal vector. As shown in Fig. 14, compared to the conventional scheme, the proposed

scheme also presents faster acquisition in fading channels.

Similar improvements can be found for the timing recovery schemes that assume

Mengali’s TEDs in MSK-type systems. These results verify that the proposed scheme

23

0 50 100 150 200 250 300 350 400 450−0.6

−0.4

−0.2

0

0.2

0.4

0.6(a) Conventional timing recovery scheme

The number of symbols

δ/T

γ=0.013

0 50 100 150 200 250 300 350 400 450−0.6

−0.4

−0.2

0

0.2

0.4

0.6(b) New timing recovery scheme

The number of symbols

δ/T

γ=0.013

Fig. 14. Performance comparison in flat fading channels, BLT = 0.01, Es/No = 20

dB, fdT = 0.0002.

24

greatly speeds up the timing acquisition.

We remark that it is not free to achieve fast time recovery. Although the same

TED can be used in the two schemes, the proposed scheme requires an additional

anti-hangup controller. Thus, the proposed scheme slightly increases the complexity

of timing synchronization. Furthermore, the original Gardner’s TED assumes only

two samples per symbol while the proposed scheme requires four samples per symbol.

It should be mentioned we do not increase the sampling requirement for Mengali’s

TED which originally assumes four samples per symbol [31].

E. Summary

In this chapter we have proposed a new anti-hangup timing recovery scheme which

assumes a two-step operation. Based on the initial timing estimate obtained in the

first step, we reduce the timing uncertainty from [−T/2, T/2] to [−T/4, T/4], and

thus avoid the hangup phenomenon in the second step. For two well-known timing

error detectors, we show through simulation results that this simple scheme greatly

speeds up the timing recovery for both linearly and nonlinearly modulated systems.

Blind feedforward timing estimator [10] was proposed to avoid hang-up problem.

However, some blind feedforward estimator present error floors at hight SNR, which

is caused by the self-noise [7]. In next chapter, a novel prefilter will be proposed to

reduce the self-noise of feedward timing estimators.

25

CHAPTER III

JITTER-FREE FEEDFORWARD TIMING RECOVERY FOR NARROWBAND

SYSTEMS

A. Jitter-Free Prefilter for Feedforward Timing Recovery of Linear Modulations

1. Introduction

Nondata-aided (blind) feedforward timing recovery schemes [10]- [15] enable fast and

reliable synchronization, and therefore, they have found use in burst transmission sys-

tems. A general cyclostationary framework that exploits second-order nonlinearities

to design digital blind feedforward synchronizers was proposed in [13].

For systems with large excess bandwidth, the performance of the synchronizers

that exploit the spectral line generated by a nonlinearity is asymptotically (large

sample) very close to the Cramer-Rao bound. If the excess bandwidth is small,

significant jitter is induced by data pattern which degrades the performance of the

timing recovery scheme in mid and high SNRs [16].

Franks and Bubrouski found that the analog second-order nonlinearity based

timing synchronizer can be jitter-free if an appropriate prefilter is used [38]. Ref-

erence [39] extended Franks and Bubrouski result to analog synchronizers that as-

sume arbitrary nonlinearities. For digital timing recovery, [40]- [41] exploited similar

prefilters to improve the performance of Gardner’s nondata aided timing recovery

scheme [30]. By means of simulation results, [42] reported that a similar prefilter

could be used to improve the performance of the four samples per symbol based feed-

forward scheme [10]. However, no rigorous theoretical analysis was conducted in [42]

to justify the jitter-free timing recovery condition.

In this section, we derive a closed-form expression for the power of self-noise

26

present in digital blind feedforward timing estimator [15], which is a modified version

of [14] and needs two samples per symbol. This rigorous derivation of the self-noise

power is later exploited for designing a prefilter that ensures nearly jitter-free timing

recovery scheme. The proposed analysis and design can be easily applied to feedfor-

ward timing recovery schemes that assume oversampling factors larger than or equal

to three (see e.g., [10]). In essence, it is shown that if the Franks-Bubrouski condition

is fulfilled, i.e., the frequency response of the equivalent pulse is symmetric with re-

spect to half the symbol rate and has bandwidth less than symbol rate, then a nearly

jitter-free digital feedforward timing recovery scheme is obtained for any oversam-

pling factor. Finally, we find the equivalence between the digital feedforward timing

estimator and the analog synchronizer can directly lead to the prefilter by utilizing

the previous results in [39].

2. Signal Model and Symbol Timing Estimators

We consider the baseband representation of a linearly modulated signal transmitted

through an AWGN channel. The receiver input is expressed as

r(t) =∑

l

alhT (t − lT − ǫT ) + w(t), (3.1)

where al stand for zero-mean unit variance (E|al|2 = 1) independently and identically

distributed (i.i.d) complex valued symbols with the fourth-order moment E[|al|4] = γ,

hT (t) is the transmitter’s filter, and w(t) is complex white Gaussian noise with two-

sided power spectral density N0/2 per component. In (1), ǫ denotes the unknown

symbol timing delay (normalized by the symbol period T ).

To simplify the analysis, we assume that the frequency offset has been compen-

sated before the timing recovery task is performed (see e.g., [13] for such frequency

compensation schemes). Since the proposed timing delay estimator is insensitive to

27

carrier phase offsets, we also omit the presence of carrier phase offset in (3.1). After

matched filtering with hR(T ), the resulting signal x(t) is oversampled by Ts := T/P ,

with the oversampling factor P ≥ 2, and the received sequence is given by

x[n] =∑

l

alhc[n − lP − ǫP ] + v[n], (3.2)

where x[n] := x(nTs), v[n] := w(t) ⊗ hR(t)|t=nTs , hc[n] := hc(nTs), hc(t) := hT (t −

ǫT )⊗hR(t), and ⊗ denotes the convolution operator. We assume that hc(t) is a raised

cosine (RC) pulse of bandwidth [−(1 + ρ)/2T, (1 + ρ)/2T ], with the roll-off factor ρ

(0 < ρ < 1) and its Fourier transform (FT) is denoted Hc(F ).

To estimate the timing offset, the second-order cyclostationary statistics of an

observation vector with N symbols will be exploited. The sample cyclic correlation

coefficient at cycle k and lag τ (integer) is given by

R̂x(k, τ) =1

NP

NP−τ−1∑

n=0

x∗[n]x[n + τ ]e−j2πkn/P , (3.3)

where ∗ denotes the conjugation operator. Based on (3.3), a general estimator was

derived by Gini and Giannakis [13] for oversampling factors P ≥ 3

ǫ̂ = − 12π

arg

Lg∑

τ=−Lg

1

G(τ)R̂x(1, |τ |)

, (3.4)

For any even and real hc(t), e.g., the RC pulse, G(τ) is a real and even function and

defined by

G(τ) :=P

T

∫ 1/2T

−1/2THc(F −

1

2T)Hc(F +

1

2T)ej2πτTF/P dF. (3.5)

The upper limit Lg of summation (3.4) is fixed as the maximum lag τ that

provides a non-zero value for |G(τ)|(≫ 0). In (3.4), Lg cyclic correlations R̂x(1, τ)

are averaged with weighting factors G−1(τ). Simulation results in [12] shows that

estimator (3.4) with small Lg exhibits good performance. Thus, in this chapter we

28

use O&M estimator (Lg = 0) for P ≥ 3. A new timing estimator using two samples

per symbol was proposed by Lee in [14]. However, Lee’s estimator is biased for large ρ

and the authors in [15] modified Lee’s estimator and obtained an unbiased estimator.

Therefore, we introduce herein the following symbol timing estimators1

ǫ̂=− 12π

arg{Γ(ǫ)} , (3.6)

Γ(ǫ)=

R̂x(1; 0) − j G(0)G(1)ℜ{R̂x(1; 1)}, P = 2

R̂x(1; 0), P ≥ 3.

The performance of the above estimators is very close to modified Cramer-Rao

bound (MCRB) [1] for large roll-off factors ρ as shown in [15] and [12]. However, for

small ρ, a large error floor will be caused by self-noise, a fact which can be observed

next section for oversampling factors P = 2. Similar results can be observed for

P ≥ 3 and are not presented here for space limitation.

To reduce this self-noise, we introduce a prefilter hpre(t) with FT Hpre(F ). Since

this chapter focuses on the compensation of self-noise, we omit the additive noise

hereafter. Thus, we replace hc[n] in (2) with h[n] := h1(nTs), where h1(t) := hc(t) ⊗

hpre(t). Also, in (3.5), Hc(F ) is replaced with H1(F ) := Hc(F )Hpre(F ), the FT of

real-valued filter h1(t), which is assumed to be an even function.

3. Nearly Jitter-Free Prefilter

At first, let us consider a special case ǫ = 0. For P = 2, considering R̂x(1; 0) is

real-valued, to get jitter-free estimation ǫ̂ = 0, we need

ℜ{R̂x(1; 1)} =1

2Nℜ{∑

l

∑

k

a∗l ak2N−2∑

n=0

h[n − 2l]h[n + 1 − 2k](−1)n}

(3.7)

1Notations ℜ{·} and ℑ{·} denote the real and imaginary part, respectively.

29

to be zero. This can be obtained if we let h[n − 2l]h[n + 1 − 2k] = 0, or equivalently

the product of two consecutive samples is zero. Since h[2n] = h1(nT ) is the data

strobe with unit amplitude, it follows that

h[2n + 1] = h1(nT + T/2)

=∫ 1/T

−1/TH1(F )e

j2πFT (n+1/2)dF

=∫ 1/T

0[H1(F ) − H1(F − 1/T )]ej2πFT (n+1/2)dF (3.8)

should be zero for any n, which results in

H1(F ) = H1(F − 1/T ) , F ∈ (0, 1/T ) . (3.9)

Considering H1(F ) is an even function, we find that, for F > 0, H1(F ) should be

symmetric around 1/2T (or −1/2T , for F < 0) and with bandwidth less than 1/T .

For RC pulse, the above result leads to the prefilter

hpre(t) = hc(t)cos(2πt/T ) . (3.10)

Actually, (3.8) was also used in [40] to derive the prefilter for Gardner’s feedback

scheme [30].

Motivated by this special result, we want to check if the above prefilter has

removed the self-noise of estimator (3.6) for general cases (ǫ 6= 0). However, for

non-zero timing offset case, the above analysis can not verify the estimator with

the prefilter (3.10) is jitter-free. Then we resort to calculate the mean-square error

(MSE) of the estimators in closed form. For simplicity, we will keep to the self-noise

calculation for P = 2.

The estimation error of (3.6) can be expressed as

ǫ̂ − ǫ = − 12π

arg{Γ(ǫ)ej2πǫ} . (3.11)

30

Taking the tan-function of both sides of (3.11), we obtain

tan[2π(ǫ̂ − ǫ)] = −ℑ{Γ(ǫ)ej2πǫ}

ℜ{Γ(ǫ)ej2πǫ} ≈ 2π(ǫ̂ − ǫ) , (3.12)

where the last approximation holds whenever the estimation error is small since

tan(x) ≈ x for |x|

31

terms involved in (3.16)- (3.17). In Fig. 15, we plot the ratios Gm(1, 0)/Gm(0, 0), m =

1, 2, for different roll-off factors ρ, where

Gm(0, V ) :=∫ 1/2T

−1/2THm2 (F, V )dF ,

Gm(1, V ) :=∫ 1/2T

−1/2THm2 (F, V )cos(πFT )dF . (3.18)

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.90.92

0.93

0.94

0.95

0.96

0.97

0.98

0.99

1

ρ

G1(1,0)/G

1(0,0)

G2(1,0)/G

2(0,0)

Fig. 15. Plots of Gm(1, 0)/Gm(0, 0) for m = 1, 2 versus various roll-off factors ρ.

Fig. 15 shows that Gm(1, 0)/Gm(0, 0) is very close to 1, especially for small ρ.

Recalling that G1(1, 0) = G(1) and G1(0, 0) = G(0), we can approximate G(1) ≈

G(0). Thus, we can approximate (3.16)-(3.17) by

Q1(V ) ≈ 2G2(0)G1(0,−V )[G1(0, V ) − G1(1, V )], (3.19)

Q2(V ) ≈ 2G2(0)[G2(0, V ) − G2(1, V )] . (3.20)

Similarly, we can also approximate Gm(1, V )/ Gm(0, V ) ≈ 1 for small V . Note

that for large V , sin2(πNV T )/(π2V 2) ≈ 0, we can approximate Q1(V ) ≈ 0 and

Q2(V ) ≈ 0 in (3.15). Thus, we obtain E[Qss] ≈ 0, which means that we can reduce

32

the self-noise with the prefilter defined in (3.10).

0 5 10 15 20 25 30 35 40 4510

−8

10−7

10−6

10−5

10−4

10−3

10−2

10−1

16QAM, ρ=0.25 N=100

Without prefilter

With prefilter

MCRB

Es/N

o (dB)

Mea

n S

quar

e E

rror

Fig. 16. Comparison of symbol timing estimators for P = 2.

To verify this result, we compare the performance of estimators with and without

prefilter for systems with low roll-off factors (ρ = 0.25) by simulations in Fig. 16. For

each SNR value, we average the performance of 1, 000 Monte-Carlo runs over different

timing offsets. At mid and high SNRs where the performance is dominated by the

self-noise, no obvious error floor is observed for SNR less than 35dB for the estimator

with prefilter and its performance is closer to the MCRB. At low SNRs where the

additive noise is dominant, the prefilter does not degrade the performance since the

same excess bandwidth is preserved. In Fig. 17, we also compare the performance of

estimator with prefilter for different N . Except for very high SNR, there is still no

error floor for short observation interval N=24, which verifies that nearly jitter-free

timing recovery can be obtained by prefilter..

The above analysis can be easily extended to O&M estimator. By calculating

self-noise in (3.6) for (P ≥ 3), we find the same filter as (3.10) can be used to remove

33

0 5 10 15 20 25 30 35 40 4510

−8

10−7

10−6

10−5

10−4

10−3

10−2

10−1

Es/N

o (dB)

Mea

n−S

quar

e E

rror

N=24, with prefilterN=24, MCRBN=100, with prefilterN=100, MCRB

16QAM, ρ=0.25, P=2

Fig. 17. Comparison of the performance of estimator for different N .

the self-noise for O&M estimator, and this theoretical result can be used to justify the

simulation results in [42]. Also, we find that this prefilter brings a new explanation

for (3.4). If we replace hpre(t) by an equivalent FIR hpre[n] of LP +1 taps and assume

R̂x(1, |τ |) are related to the samples at the input of the prefilter, the timing estimator

(3.6) for P ≥ 3 can be viewed as

ǫ̂ = − 12π

arg

LP∑

τ=−LPrhpre [τ ]R̂x(1, |τ |)

, (3.21)

where rhpre [τ ] :=∑LP +|τ |

τ1=|τ | hpre[τ1]h∗pre[τ1 − |τ |] can be viewed as the autocorrelation

coefficient at lag τ of the impulse response of the prefilter. Thus, to remove the

self-noise, the 2Lp + 1 cyclic correlations in (3.4) should be averaged with weighting

factors that depend on the prefilter’s autocorrelation coefficients.

We remark that in frequency-selective channel the symmetry of H1(F ) can not

be obtained unless the channel is perfectly known. Thus, jitter free timing recovery

cannot be obtained by simple prefiltering in this channel.

34

4. Summary

We have derived a closed-form expression for the power of self-noise for a blind feed-

forward symbol timing estimators that assume oversampling factors equal to 2. It

has been shown that an appropriate prefilter after the receiver’s matched filter can

be utilized to reduce the jitter. This result can be easily extended for estimators with

higher (P ≥ 3) oversampling factors. Simulation results prove that the synchronizer

which assumes such a prefilter is nearly jitter-free even in the presence of a finite

(reduced) number of samples as long as linear-modulated signals are assumed. Such

prefilters actually can improve performance of square-law estimators for MSK and

GMSK-type signals, a topic which will be researched in the next section.

B. Jitter-Free Prefilter for Feedforward Timing Recovery of GMSK Modulations

1. Introduction

According to Laurent’s approximation [43], minimum-shift keying (MSK)-type signals

can be viewed as a superposition of some linearly modulated waveforms. Based on this

interpretation, several blind feedforward timing estimators, similar in structure to the

schemes proposed for linear modulations [10]- [15], were proposed for MSK/Gaussian

MSK (GMSK) modulations in [44]- [47]. At low signal-to-noise ratios (SNRs), the

estimators [46]- [47] approach the Cramer-Rao bound (CRB) and outperform the

timing recovery schemes [44]- [45]. By exploiting jointly the information contained

in the in-phase and quadrature components of the received signal, a novel timing

recovery algorithm with improved performance relative to the timing estimator [46] is

proposed. Unfortunately, due to self-noise (jitter), the performance of timing recovery

schemes [46] and [47] for GMSK modulations is far from CRB at mid and high SNRs.

Motivated by the fact that prefilter based structures can improve the performance of

35

feedforward timing estimators for linear modulations (see [48] and the references cited

therein), this chapter proposes a similar prefilter to reduce the self-noise of timing

recovery schemes [46]- [47]. Computer simulations show that the proposed prefilter

based timing recovery schemes exhibit improved performance at mid and high SNRs

with respect to the existing schemes [46]- [47].

2. Signal Model and Estimators

According to Laurent’s approximation [43], an MSK-type signal can be approximated

by

s(t) ≈∑

l

exp [jπ

2

l∑

i=1

Ii]hT (t − lT ) , (3.22)

where

hT (t) :=L−1∏

i=0

p(t + iT ) , (3.23)

p(t) :=

sin[πq(t)], 0 ≤ t ≤ LT

p(2LT − t), LT < t ≤ 2LT

0, otherwise.

(3.24)

In (3.22)-(3.24), T denotes the symbol period, Il stands for the zero mean and in-

dependently and identically distributed (i.i.d) binary information sequence and q(t)

is the phase response of the modulator, supposed of length L. If the signal s(t) is

assumed to be transmitted through an AWGN channel, after carrier frequency and

phase offset compensation, the receiver output can be expressed as

r(t) = s(t − ǫT ) + w(t), (3.25)

where w(t) stands for complex white Gaussian noise with two-sided power spectral

density N0/2 and ǫT denotes the unknown symbol timing delay. Since the MSK-

36

type signal can be viewed as a linearly modulated signal with pulse shaping filter

hT (t), the received signal is processed through the matched filter hR(t) := hT (−t)

and oversampled with the sampling period Ts = T/P (P ≥ 3). Consequently, the

received sequence can be expressed as

x(n) =∑

l

alh(n − lP − ǫP ) + v(n) , (3.26)

where al ∈ {±j} for odd l, al ∈ {±1} for even l, v(n) := w(t) ⊗ hR(t)|t=nTs, h(n) :=

hT (t)⊗hR(t)|t=nTs, and ⊗ denotes the convolution operator. Also, let H(f) stand for

the Fourier transform (FT) of h(n). After performing some approximations (accurate

only for low SNRs) on the log-likelihood function, Morelli and Vitetta derived the

blind feedforward timing estimator [47]

ǫ̂ := − 12π

arg

{P−1∑

τ=0

|R(τ)|2e−j2πτ/P}

, (3.27)

where R(τ) is obtained by an operation of squaring the observation vector of length

L0:

R(τ) =L0−1∑

n=0

(−1)n+1x2(nP + τ) , (3.28)

and the modulating factor (−1)n+1 is used to remove the time-varying effects intro-

duced by the data modulation. The above estimator is very similar to the digital

timing recovery algorithm proposed for linear modulations in [10].

A feedforward clock recovery configuration is shown in Fig. 18. The estimated

value ǫ̂ is post-processed by a lowpass filter with saw-tooth nonlinearity [10] to con-

trol the interpolator. The prefilter hpre(n) will be later introduced to improve the

performance of timing estimator by reducing its self-noise.

Note that (3.26) can be rewritten as

x(n) =∑

l

b2lh(n − 2lP − ǫIP ) + j∑

l

b2l+1h(n − 2lP − ǫQP ) + v(n) , (3.29)

37

FixedClock ~

Timing Estimator

Matched

To

DetectorInterpolator

Filter Prefilterx[n]Postprocessor

r(t)

Fig. 18. Feedforward clock recovery scheme.

where bl ∈ {±1}, ǫI = ǫ and ǫQ = ǫ + 1. Thus, an MSK-type signal can be viewed

as an offset QPSK (OQPSK) modulated signal and both in-phase (I) and quadrature

(Q) components of (3.29) are cyclostationary with period 2P . Following the steps

in [13], the cyclic correlations of the real and imaginary components of x(n) are found

to be

∑

n

{ℜ[x(n)]}2e−j2πn/2P ∝ e−jπǫI ,∑

n

{ℑ[x(n)]}2e−j2πn/2P ∝ e−jπǫQ , (3.30)

where ℜ[x] and ℑ[x] denote the real and imaginary parts of the complex-valued num-

ber x, respectively. In [46], estimates of the timing delay are obtained by exploiting

separately the in-phase (I) and quadrature (Q) components of the received signal,

and are given by

ǫ̂I = −1

πarg

L0P−1∑

n=0

{ℜ[x(n)]}2e−j2πn/2P

,

ǫ̂Q = −1

πarg

L0P−1∑

n=0

{ℑ[x(n)]}2e−j2πn/2P

. (3.31)

Estimator [46] exploits separately these estimates to obtain an estimate of the timing

38

delay ǫ. However, we notice that if both I and Q components are jointly exploited

to estimate ǫ, the performance of [46] can be further improved. Taking (3.30) into

account, this new estimator takes the form:

ǫ̂ := − 1π

arg

L0P−1∑

n=0

{ℜ[x(n)]}2e−j2πn/2P + ejπ ·L0P−1∑

n=0

{ℑ[x(n)]}2e−j2πn/2P

,

= − 1π

arg

L0P−1∑

n=0

ℜ[x2(n)]e−jπn/P

. (3.32)

0 5 10 15 20 25 30 35 40 4510

−7

10−6

10−5

10−4

10−3

10−2

10−1

WN estimator (I only)WN estimator (Q only)New estimator MV estimatorMCRB

Nor

mal

ized

Mea

n S

quar

e E

rror

Eb/N

o (dB)

L=100 GMSK, BT=0.3

Fig. 19. Performance comparisons of various estimators for GMSK modulations.

In Fig. 19, computer simulations illustrate the performance of the above men-

tioned estimators for GMSK modulations. The performance of the new estimator

(3.32) is always better than that corresponding to the estimator that relies only on

either the I or Q-component [46]. If no phase offset is assumed, the performance

of (3.32) is even slightly better than than that exhibited by the Morelli and Vitetta

(MV) estimator. Unfortunately, (3.32) is sensitive to residual phase offsets while the

MV estimator works well for large phase offsets. Because the prefilter to be discussed

39

in next section provides more improvement for GMSK modulations rather than for

MSK modulations, in what follows we will focus only on designing prefilter based

timing estimators for GMSK modulations.

3. Self-Noise Compensating Prefilter

At low SNRs, the performance of MV estimator is very close to CRB since it is

an approximate ML estimator at low SNRs. However, the MV estimator is not

necessarily good at medium and high SNRs since its approximation is not accurate

anymore. It is known that the self-noise dominates the performance of such estimators

at high SNRs. The simulation results presented in Fig. 19 show that for GMSK

modulations the performance of the MV estimator is far away from CRB at high

SNRs, which indicates that large self-noise exists in (3.27).

For linear modulations, some prefilters have been found to reduce the self-noise in

analog synchronizers [38] as well as in digital synchronizers [40] and [48] that exploit

second-order nonlinearities. Since the MV estimator is very similar to the second-

order nonlinearity based timing recovery algorithm [10] 2, we expect that a properly

designed prefilter can also reduce the self-noise of both the MV estimator and the

new estimator (3.32), which is also similar in structure to [10].

Both estimators (3.27) and (3.32) exploit two key facts: an MSK-type signal is

equivalent to an OQPSK modulation, and the symbol period of the I and Q compo-

nents of an MSK-type signal is 2T . Therefore, we find that the excess bandwidth of

MSK-type signals is the part of signal bandwidth larger than 1/4T instead of 1/2T

(which holds for linear modulations). Thus, to reduce the self-noise, we introduce

2The only difference is that for removing the effect of possible phase offsets in(3.27), the absolute value of R(τ) is computed before evaluating the complex Fouriercoefficients.

40

a prefilter similar in structure to the one proposed in [48] for feedforward timing

recovery in linear modulations

hpre(n) := cos(πn/P )h(n) . (3.33)

In the frequency domain, (3.33) is just the shifted version (shifted by 1/2T ) of h(n)

such that the FT of hpre(n) ⊗ h(n), Hpre(f) · H(f), is almost3 symmetrical around

1/4T .

0 0.5 1 1.5 20

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1|H(f)||H

pre(f)|

|H(f)Hpre

(f)|

f/T

Am

plitu

de R

espo

nse

Fig. 20. Amplitude response of filters.

This prefilter is shown in Fig. 20 for GMSK modulations. This prefilter is ex-

pected to reduce the self-noise of timing recovery estimators. Similar results have

been reported and analytically proven in [48] for linear modulations. To investi-

gate the performance of prefilter based timing recovery estimators, simulation results

for GMSK modulations with pre-modulator bandwidth BT = 0.3 are illustrated in

3To maintain Hpre(f)·H(f) strictly symmetrical around 1T , an additional bandpassfilter may be required. However, for GMSK, we find that (3.33) is sufficient to obtainthe desired symmetry.

41

Fig. 21. The observation length is 100 and the oversampling factor is P = 4. All

simulation results are obtained by performing 2,000 Monte-Carlo trials averaged over

different timing offsets for each SNR value. Also, the modified CRB (MCRB) [1] is

plotted.

0 5 10 15 20 25 30 35 40 4510

−7

10−6

10−5

10−4

10−3

10−2

10−1

MV estimator, without prefilterMV estimator, with prefilterNew estimator, without prefilterNew estimator, with prefilter MCRB

Eb/N

o (dB)

Nor

mal

ized

Mea

n S

quar

e E

rror

Fig. 21. The performance of estimators with and without prefilter.

MCRB(ǫ) =1

2π2 · L0 · Eb/No ·∫∞−∞ Tg

2(t)dt, (3.34)

where g(t) is the frequency response of modulator. The simulation results indicate

that the proposed prefilter improves significantly the performance of the MV estimator

and the new estimator (3.32) for SNR values equal to and larger than 15 dB, while

at low SNRs their performance is almost the same. Also, the proposed prefilter

can work for MSK modulations (a result that is not detailed herein). Since the

excess bandwidth of GMSK modulations is much smaller than that corresponding to

MSK modulations, much more significant self-noise is induced in (3.27) and (3.32).

Therefore, the usage of self-noise compensating prefilters leads to larger performance

42

improvement for GMSK modulations. We notice further that there is still an error

floor at very high SNRs, a fact which can be explained by the Laurent approximation

error in (3.22) and the approximation made in [48] when deriving the expression of

self-noise. As a side observation, we remark that the above mentioned improvement

is obtained by increasing the complexity of the receiver since an additional prefilter

besides the matched filter HR(t) is required. However, we have found that a prefilter

with 20 taps is precise enough for GMSK modulations with BT=0.3.

4. Summary

A new timing estimator with improved performance relative to the Wu and Ng esti-

mator [46] has been proposed for MSK-type modulations. The performance of this

new estimator is slightly better than that of Morelli and Vitetta estimator [47]. How-

ever, both estimators are sensitive to phase offset errors. We have also derived a

new digital prefilter to reduce the self-noise for both the MV estimator [47] and the

WN estimator [46], a result which represents an extension of [48]. Simulation results

assessed in AWGN channels show that the estimators with the proposed prefilter are

very close to CRB for practical SNR. For GMSK modulations with pre-modulator

bandwidth of 0.3, this simple prefilter improves the performance of existing timing

recovery schemes for SNRs equal to or larger than 15 dB. The proposed prefilter

based timing recovery estimators are almost jitter free and present slightly increased

complexity.

43

CHAPTER IV

EFFICIENT COARSE FRAME AND CARRIER SYNCHRONIZATION FOR

OFDM SYSTEMS

A. Introduction

In last two chapters, we have discussed timing recovery for single carrier modulated

systems, where we assume AWGN or flat fading channel. As the system bandwidth

increases, the propagation channel becomes more frequency selective. Due to its high

complexity and slow convergence, the conventional equalizer (time domain) is not fit

for wideband systems. Thanks to the simple frequency domain equalization, orthog-

onal frequency division multiplex (OFDM) systems have been adopted in wideband

systems.

However, due to the sensitivity of OFDM systems to synchronization errors (espe-

cially to carrier frequency offset), reliable synchronization schemes must be designed

for these systems. By exploiting the known structure of a training symbol or a cyclic

prefix, several schemes have been proposed for coarse estimation of the carrier fre-

quency offset and timing delay [18]- [26], and [49].

In [19], the training symbol contains two identical halves [+A +A], and the

timing delay estimator is obtained by searching for the peak of correlation between

the first and second halves of the received symbols. By comparing the phase difference

between the identical parts, a coarse frequency offset estimator is also proposed.

However, the correlation peak of the timing metric exhibits a plateau which causes

large variance for the timing estimator [21]. Based on a training preamble having

the same structure as the one used in the Schmidl-Cox’s estimator [19], reference [20]

showed that the performance of the MMSE (Minimum Mean-Square Error) and ML

44

(Maximum Likelihood) timing estimators still perform unsatisfactorily.

Similar to the signaling set-up adopted in [19], Coulson used two repeated m-

sequences as a training symbol [24]- [25]. However, the proposed time synchronization

algorithm is likely to fail in the presence of large carrier frequency offsets, and presents

high implementation complexity due to the matched filtering. We remark that a

training symbol with the same structure as the one proposed in [24] is exploited

in [26] to develop reliable frequency and time acquisition schemes. However, the

proposed time synchronization algorithm is also sensitive to large frequency offsets.

By adopting a structured training symbol of the form [+B +B -B -B], Bhargava

et.al. proposed a coarse timing delay estimator that outperformed Schmidl-Cox’s es-

timator [21]. However, reference [21] does not provide any detailed insight or analysis

pertaining to the features of this estimator. Inspired by the signaling set-up proposed

in [21], this chapter aims to develop reliable and reduced complexity coarse frame

and carrier frequency acquisition schemes that exploit a structured training symbol

of the form:

[±B ± B ± B ± B] , (4.1)

where B stands for a sequence of N/4 training samples with constant variance (power)

(e.g., an m-sequence), and it can be generated with good approximation by using an

N/4-point IFFT of an m-sequence. Principally, any signal with constant envelope in

the time domain and a bandwidth similar to the OFDM data symbol can be used as

a training symbol. It is found that among all the signaling set-ups (4.1), the training

symbol:

[+B + B − B + B] , (4.2)

leads to timing acquisition schemes that exhibit the best detection properties in terms

of lower false detection probability and higher correct acquisition probability. In

45

addition, it turns out that the negative term -B present in (4.2) can be placed in

any location with no loss in performance1. By exploiting the structured training

symbol (4.2), this chapter proposes robust acquisition schemes for time delay and

carrier frequency offset for OFDM systems that operate in additive white Gaussian

noise (AWGN) and frequency-selective channels. The proposed time synchronizer

offers more accurate estimates than the estimators [19] and [21]. It is also found that

the performance of the proposed time and frequency offset estimators is nearly the

same as [24]. In addition, the proposed estimator requires a reduced implementation

complexity and is more robust to large frequency offsets with respect to (w.r.t) [24].

The rest of chapter is organized as follows. In Section B, we describe the signal

model and introduce some modeling assumptions. In Section C, an optimum maxi-

mum likelihood (ML) estimator is derived for the continuous transmission scenario.

We modify this ML estimator to a sub-optimum estimator with reduced complex-

ity, which is shown to exhibit robust performance to fading channel for both burst

and continuous transmission scenarios. A theoretical performance analysis study is

conducted in Section D. Finally, Section E describes computer simulations that illus-

trate the advantages of the proposed estimator and that corroborate the theoretical

performance analysis performed in Section D.

B. Signal Model

The OFDM baseband signal is generated by the IFFT-transform:

x(n) =1√N

N−1∑

k=0

skej2πkn/N , −L ≤ n ≤ N − 1 , (4.3)

1All of them are similar to the bark code with length 4

46

where sk represents the data sequence modulated on the kth subcarrier, which may

assume any modulation format (such as QAM or PSK), and is independently and

identically distributed (i.i.d.) with zero mean and variance E{|sk|2} = σ2s . Let N ,

L, and 1/Tu = 1/(NTs) denote the number of subcarriers, the length of cyclic prefix

(guard time), and the subcarrier spacing, respectively. Normally, the length of cyclic

prefix L is selected to be not more than N/5, which can be interpreted as a 1-dB

signal-to-noise ratio (SNR) loss introduced by the cyclic prefix [17, p. 46]. Herein,

without loss of generality (w.l.o.g.) the value L = N/8 is adopted.

At the beginning of our study, we assume a flat fading channel model to derive the

maximum likelihood estimator. Later it will be shown by computer simulations that

the proposed estimator works well for frequency-selective channels, too. Assuming

the sampling period Ts = Tu/N , the received signal samples can be expressed as:

r(n) = α(n)x(n − ϑ)ej(2πfnn/N+θ) + w(n) , (4.4)

where ϑ stands for the timing offset, α(n) is the channel amplitude, fn := feTu =

feNTs is the normalized carrier frequency offset, θ denotes the phase offset, and w(n)

denotes the samples of a zero-mean complex white Gaussian noise random process

with variance E{|w(n)|2} = σ2w and is assumed independent w.r.t x(n). In slow-

varying channels, we can assume α(n) to be a constant α over the duration of several

OFDM symbols. The signal-to-noise ratio is represented in terms of the variable

SNR:=α2σ2s/σ2w.

After coarse frame and carrier synchronization is achieved, the receiver discards

the cyclic prefix, and the modulated symbol stream {sk} can be recovered by means

of an FFT-operation. Due to the presence of cyclic prefix, small (fractional) time

offsets ϑ in magnitude less than the interval (L) spanned by the cyclic prefix cause no

ISI or interchannel interference (ICI) [49]- [50]. The time offset induces a phase offset

47

exp (−j2πϑn/N) on the nth subcarrier, which can be corrected using channel esti-

mation techniques. Therefore it is enough to estimate the start of training sequence

within one sample period.

This chapter focuses on the estimation of fn (packet detection) and ϑ. Although,

α and θ are unknown to the receiver, their estimation can be avoided via differential

encoding/decoding or can be obtained wi